Logical Logarithms

Algebra Level 3

logabc=logbca=logcab\large { \frac{\log {\color{#D61F06}a}}{{\color{#20A900}b}-{\color{#3D99F6}c}}} {\color{#624F41}=}\frac{\log{ \color{#20A900}b}}{{\color{#3D99F6}c}- {\color{#D61F06}a}} {\color{#624F41}=}\frac{\log {\color{#3D99F6}c}}{{\color{#D61F06}a}-{\color{#20A900}b}}

a,b,c a, b, c are distinct positive real numbers that satisfy the above equation. What is

aa×bb×cc= ?\large {{\color{#D61F06}a}^{\color{#D61F06}a} \times {\color{#20A900}b}^{\color{#20A900}b} \times {\color{#3D99F6}c}^{\color{#3D99F6}c}} = \ ?

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